I'm asking this question because of an article in New Scientist about a recent preprint by a group including Lee Smolin. I haven't taken the time to comprehend the paper completely. My knowledge of differential and pseudo-Riemannian geometry is out of practice, so it'd take me a while to do so. Even so, the paper put some ideas in my head about phase space and symplectic manifolds that can be put as simple questions but I couldn't answer with some basic Googling. Also, this is resting on the (mis)understanding that phase space is a symplectic manifold. If this is wrong, these questions might be meaningless.
Is there a canonical symplectic manifold that is associated with a given pseudo-Riemannian manifold? I'm guessing yes, because a pseudo-Riemannian manifold is itself a differential manifold, and I think the tangent bundle is a symplectic manifold.
If the space we started with has a metric, does this imbue the associated symplectic manifold with any structure? e.g. is the symplectic form limited by or related to the metric or something similar?
Could any ideas from SR/GR be extended to this symplectic space such that we learn something new? e.g. could there be something like the Einstein Field Equations (EFEs), written for the 8-dimensional phase space, that have new solutions that are realistic but wouldn't be found using the EFEs? Or could there be non-canonical symplectic manifolds associated with a given pseudo-Riemannian space? (I suspect Darboux's theorem says "no".)
This is mostly crazy speculation based on knowledge I haven't exercised in a few years now. For all I know, there's some underlying concept that makes this pointless or I've just totally misunderstood the paper in the first place. Also, I've posted here because of the theoretical physics nature of the problem, but I accept it may be better served on another SE.